unitary matrix

Question

A is a unitary matrix. Then eigen value of A are
(a) 1, – 1 (b) 1, – i
(c) i, – i (d) – 1, i

Comments

$A$ is  the unity matrix, iff $A^{\theta} = A^{-1}$

$\implies A\cdot A^{\theta} = A\cdot A^{-1} $

$\implies A\cdot A^{\theta} = I $

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I think that the answer is C.

beacuse for a unitary matric determinant is equal to 1.

so i*(-i)=1

Answer 1

Unitary Matrix: If matrix  A is called Unitary matrix then it satisfy this condition  $A.A^{\theta } = A^{\theta }.A = I$

where  $A^{\theta } = $ Transpose Conjugate of $A = (A')^{T}$ (first you Conjugate andthen Transpose , you will get Unitary matrix)

Properties of Unitary matrix:

1. If $A$ is a Unitary matrix then$A^{-1}$ is also a Unitary matrix.
2. If $A$ is a Unitary matrix then  $A^{\theta }$ is also a Unitary matrix.
3. If $A \& B$ are Unitary matrices, then $A.B$ is a Unitary matrix.
4. If $A$ is Unitary matrix then  $A^{-1} = A^{\theta }$
5. If $A$ is Unitary matrix then it's determinant is of Modulus Unity $(\text{always}\: 1).$

Let A = $\begin{bmatrix} \frac{1+i}{2} &\frac{-i+1}{2} \\ \frac{1+i}{2} &\frac{1-i}{2} \end{bmatrix}$ is Unitary matrix

Characteristics equation  of matrix $A$ is $ \mid A-\lambda I\mid = 0$

$\begin{vmatrix} \frac{1+i-2\lambda }{2} &\frac{-1+i}{2} \\ \frac{1+i}{2} &\frac{1-i-2\lambda }{2} \end{vmatrix} = 0$

$\implies(\frac{1+i-2\lambda }{2}) (\frac{1-i-2\lambda }{2}) - (\frac{1+i}{2}) (\frac{i-1}{2})=0$

$\frac{4\lambda ^{2} -4\lambda + 2}{4} - \frac{(-2)}{4} = 0$

$\implies4\lambda ^{2} -4\lambda + 4 = 0$

$\implies4(\lambda ^{2} -\lambda +1) = 0$

$\implies\lambda ^{2} -\lambda +1 = 0$

$\lambda = \frac{-(-1)\pm\sqrt{(-1)^{2}-4(1)(1)} }{2}$

$\implies\lambda = \frac{1\pm\sqrt{1-4} }{2}$

$\implies\lambda = \frac{1\pm\sqrt{-3} }{2}$

$\implies\lambda = \frac{1\pm\sqrt{-1}.\sqrt{3} }{2}$

$\implies\lambda = \frac{1 + \sqrt{3}.i }{2}$ , $\frac{1 - \sqrt{3}.i }{2}$

Comments

@Lakshman Patel RJIT

How u suppose matrix A??

is it one possible matrix??

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@Hradesh patel

this i took from the book, but now I didn't remember the book name